In Sect.  we studied dispersion
relations for the graviton amplitudes of various helicities. The key
observation is that we can apply dispersion relations to matrices (REF ), composed of the different helicity amplitudes, whose
discontinuity is positive semi-definite. Applying dispersive sum
rules along the lines of Ref. [1]} to these matrices
we derived constraints on the Wilson coefficients which appear in the
single-minus and all-plus gravitational scattering amplitudes. Notably
we placed a bound on the \(R^3\)  coefficient that corrects the graviton
three-point amplitude in terms of the \(R^4\)  coefficient, making a step
towards making the analysis of Ref. [2]} quantitatively
precise.
